A small anecdote: my dad is a mathematician. For a significant portion of his postdoc/early career (in the 80's/90's) he worked on proving a particular conjecture. Eventually he abandoned it and went to be much more successful in other areas.

A few years ago someone found a counterexample. He was quite depressed for a few weeks at the thought of how much of his strongest research years had been devoted to something impossible.

Choosing a "good first problem" in math is quite difficult. It needs to be "novel," somewhat accessible, and possible to solve (which is an unknown when you're starting out)!

At least he didn't "prove" a theorem that turned out to be false!

Thanks that is a good anecdote. Did he get over it and how?

To me such a career is useful for (a) the greater good: you can't make discoveries without dead ends and (b) the maths created along the way! Or if not shares then the skills developed.

At least, we can assume that after a few weeks, he became not quite that depressed.